On non-uniqueness issues related to the permeability of a porous medium with a random porous structure

TL;DR

This paper investigates the non-uniqueness of permeability in porous media with random structures, revealing significant variability and discussing fundamental issues in defining and reconstructing transport properties.

Contribution

It demonstrates permeability variability across large ensembles, critiques conventional transport assumptions, and highlights the ill-posed nature of reconstructing porous structures from macroscopic data.

Findings

Permeability varies up to two orders of magnitude in low porosity media.

Permeability histograms do not narrow with larger domain sizes.

Conventional advective transport assumptions are inadequate and ill-posed.

Abstract

This paper discusses the issue of non-uniqueness of the permeability of a porous medium with a random structure. The permeability range for 12,000 realizations of a random porous structure is examined using a recently-developed modelling approach, which results in up to two orders of magnitude permeability variations in low porosities. These findings are compared with previous results for 13,000 realizations with a scaled-down size, and it is shown that the permeability histogram does not converge towards a narrower spectrum using larger domain sizes. The similarity between advective transport in an ensemble of porous media and a random walk phenomenon, accepted in the literature, is revisited and the inadequacy of the assumptions employed is discussed. It is shown that the conventional definition of advective transport in a porous medium, which is generally assumed to follow Hadamard's definition of well-posedness, cannot be realised. The reconstruction of a porous structure from macroscopic parameters is in itself an ill-posed inverse problem. To clarify this issue, explaining the ontic and epistemic identifications of a porous medium, it is discussed that while ontic identification cannot be employed for transport analysis, epistemic identification leads to non-unique solutions. It is finally suggested that a paradigm shift is required for better formulation of the transport characteristics of porous media.